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Shock Acceleration Model for the Toothbrush Radio Relic [HEAP]

Although many of the observed properties of giant radio relics detected in the outskirts of galaxy clusters can be explained by relativistic electrons accelerated at merger-driven shocks, significant puzzles remain. In the case of the so-called Toothbrush relic, the shock Mach number estimated from X-ray observations ($M_{\rm X}\approx 1.2-1.5$) is substantially weaker than that inferred from the radio spectral index ($M_{\rm rad}\approx 2.8$). Toward understanding such a discrepancy, we here consider the following diffusive shock acceleration (DSA) models: (1) {\it weak-shock models} with $M_s \lesssim 2$ and a pre-existing population of cosmic-ray electrons (CRe) with flat energy spectrum, and (2) {\it strong-shock models} with $M_s \approx 3$ and either shock-generated supra-thermal electrons or pre-existing fossil CRe. We calculate the synchrotron emission from the accelerated CRe, following the time evolution of the electron DSA, and subsequent radiative cooling and postshock turbulent acceleration. We find that both models could reproduce reasonably well the observed integrated radio spectrum of the Toothbrush relic, but the observed broad transverse profile requires the stochastic acceleration by downstream turbulence. Moreover, to account for the almost uniform radio spectral index profile along the length of the relic, the weak-shock models require a preshock region over 400 kpc with uniform population of pre-existing CRe with high cutoff energy ($\gtrsim 40$ GeV). Due to the short cooling time, it is challenging to explain the origin of such fossil electrons. Therefore, we suggest the strong-shock models with low-energy seed CRe ($\lesssim 150$ MeV) are preferred for the radio observations of this relic.